Preface |
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CHAPTER I. THE BASIC CONCEPTS |
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1 | (64) |
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1 | (2) |
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2. Compact and finitely compact metric spaces |
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3 | (7) |
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3. Convergence of point sets |
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10 | (4) |
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14 | (5) |
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5. Curves and their lengths |
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19 | (8) |
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27 | (3) |
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30 | (6) |
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36 | (8) |
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9. Multiplicity. Geodesics without multiple points |
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44 | (5) |
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10. Two-dimensional G-spaces |
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49 | (7) |
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11. Plane metrics without conjugate points |
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56 | (9) |
CHAPTER II. DESARGUESIAN SPACES |
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65 | (50) |
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65 | (1) |
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13. Planes with the Desargues property |
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66 | (10) |
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14. Spaces which contain planes |
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76 | (6) |
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15. Riemann and Finsler spaces. Beltrami's theorem |
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82 | (5) |
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16. Convex sets in affine space |
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87 | (7) |
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94 | (11) |
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105 | (10) |
CHAPTER III. PERPENDICULARS AND PARALLELS |
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115 | (50) |
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115 | (2) |
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20. Convexity of spheres and perpendicularity |
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117 | (7) |
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21. Characterization of the higher-dimensional elliptic geometry |
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124 | (6) |
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22. Limit spheres and co-rays in G-spaces |
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130 | (7) |
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23. Asymptotes and parallels in straight spaces |
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137 | (7) |
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24. Characterizations of the higher-dimensional Minkowskian geometry |
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144 | (9) |
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25. Characterization of the Minkowski plane |
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153 | (12) |
CHAPTER 1V. COVERING SPACES |
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165 | (70) |
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165 | (2) |
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27. Locally isometric spaces |
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167 | (7) |
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28. The universal covering space |
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174 | (7) |
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181 | (7) |
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30. Locally Minkowskian, hyperbolic, or spherical spaces |
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188 | (11) |
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31. Spaces in which two points determine a geodesic |
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199 | (5) |
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32. Free homotopy and closed geodesics |
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204 | (11) |
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33. Metrics without conjugate points on the torus |
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215 | (8) |
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34. Transitive geodesics on surfaces of higher genus |
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223 | (12) |
CHAPTER V. THE INFLUENCE OF THE SIGN OF THE CURVATURE ON THE GEODESICS |
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235 | (72) |
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235 | (2) |
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237 | (11) |
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37. Non-positive curvature in the theory of parallels |
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248 | (6) |
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38. Straightness of the universal covering space r |
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254 | (4) |
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39. The. fundamental groups of spaces with convex capsules |
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258 | (4) |
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40. Geodesics in spaces with negative curvature |
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262 | (5) |
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41. Relation to non-positive curvature in standard sense |
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267 | (6) |
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273 | (9) |
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43. Excess and characteristic |
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282 | (10) |
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44. Simple monogons, total excess, surfaces with positive excess |
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292 | (15) |
CHAPTER VI. HOMOGENEOUS SPACES |
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307 | (96) |
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307 | (2) |
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46. Spaces with flat bisectors I |
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309 | (11) |
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47. Spaces with flat bisectors II |
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320 | (13) |
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48. Applications of the bisector theorem. The Helmholtz-Lie Problem |
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333 | (10) |
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343 | (7) |
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50. New characterizations of the Minkowskian spaces |
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350 | (9) |
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51. Translations along two lines |
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359 | (7) |
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52. Surfaces with transitive groups of motions |
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366 | (8) |
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53. The hermitian elliptic and similar spaces |
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374 | (11) |
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54. Compact spaces with pairwise transitive groups of motions |
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385 | (9) |
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55. Odd-dimensional spaces with pairwise transitive groups of motions |
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394 | (9) |
APPENDIX. Problems and Theorems |
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403 | (10) |
NOTES TO THE TEXT |
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413 | (4) |
INDEX |
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417 | |